Find the value of $ \int \limits _0^{\infty} \large\frac{\tan^{-1} (ax)}{x(1+x^2)}$$dx$

Question

\[\begin {array} {1 1} (a)\;\frac{\pi}{2} \ln (1+a) \\ (b)\;\frac{\pi}{4} \tan ^{-1} (1+a) \\ (c)\;\frac{\pi}{2} \tan ^{-1} (1+a) \\ (d)\;\frac{\pi}{4} \log (1+a) \end {array}\]